Is it okay to leave my chronograph function running all the time?

On most mechanical chronographs, yes it is okay. Some people prefer to do this in order to have the large second hand running. There is no evidence this causes any additional wear of any significance to the watch. It also makes little difference in the power reserve on your mechanical watch. Omega reports that their watches should lose less than 2% of their power reserve having the chronograph running continuously.

On quartz chronographs it is certainly okay. Though some quartz chronographs may limit the duration that the chronograph will run. Also, some quartz watches may disable the chronograph functions to conserve battery power when the battery is very low.